Normal Distributions and Rescaling

class: center, middle, inverse, title-slide

.title[
# Normal Distributions and Rescaling
]
.author[
### S. Mason Garrison
]

---

layout: true
  
<div class="my-footer">
<span>
<a href="https://psychmethods.github.io/coursenotes/" target="_blank">Methods in Psychological Research</a>
</span>
</div>

---

class: middle

# Normal Distribution

---

## Normal Distribution

Def: a particular bell-shaped curve that has the following mathematical properties

`$f(x)= \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$`

- Formula has two parameters
    - `$\mu$`
    - `$\sigma$`
- The standard normal `$(\mu=0; \sigma=1)$` simplifies the equation

---

# Standard Normal with multiple means

- The mean is located at the center of the symmetric curve and is the same as the median. 
- Changing `$\mu$` without changing `$\sigma$` moves the Normal curve along the horizontal axis without changing its variability.

.small[
<img src="data:image/png;base64,#rnorms_files/figure-html/norm-1.png" width="55%" style="display: block; margin: auto;" />
]

---

.small[

``` r
#### Normal Distribution

# Display the normal distributions with various means

x <- seq(-80, 80, length=1000)
hx <- dnorm(x)
colors <- c("red", "blue","green", "green", "gold", "black")

plot(x, hx, type="l", lty=2, xlab="x value",
		 ylab="Density", main="Comparison of Normal Distributions",xlim=c(-5, 7))
location<-c(2,4,-2)
for (i in 1:3){
	lines(x, dnorm(x,mean=location[i]), lwd=1, col=colors[i])
}
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-2-1.png" width="40%" style="display: block; margin: auto;" />
]

---

# Plots of the Standard Normal with multiple standard deviations
.pull-left[
- The standard deviation `$\sigma$` controls the variability of a Normal curve. 
- When the standard deviation is larger, the area under the normal curve is less concentrated about the mean.
- The standard deviation is the distance from the center to the change-of-curvature points on either side.
]
.pull-right.small[
<img src="data:image/png;base64,#rnorms_files/figure-html/zsd-1.png" width="90%" style="display: block; margin: auto;" />
]
---

.small[

``` r
# Display the normal distributions with various standard deviations
plot(x, hx, type="l", lty=2, xlab="x value",
		 ylab = "Density", main="Comparison of Normal Distributions",xlim=c(-10, 10))

for (i in c(.5,2,4,6)){
	lines(x, dnorm(x,sd=i), lwd=1, col=colors[i])
}
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-3-1.png" width="40%" style="display: block; margin: auto;" />
]
---

# Normal Distribution

.pull-left[
- In the Normal distribution, with mean `$\mu$`  and standard deviation `$\sigma$`:
    - approximately `$68\%$` of the observations fall within 1 `$\sigma$` of `$\mu$`
    - approximately `$95\%$` of the observations fall within 2 `$\sigma$` of `$\mu$`
    - approximately `$99.7\%$` of the observations fall within 3 `$\sigma$` of `$\mu$`
- This property is sometimes called: The `68-95-99.7 Rule`
]
.pull-right[
<img src="data:image/png;base64,#../img/normal.png" width="95%" style="display: block; margin: auto;" />
]

---

---

# Worked Example

.pull-left[
- The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for seventh-grade students in Gary, Indiana, is close to Normal. 
- Suppose the distribution is `N(6.84, 1.55)`.
]
--

.pull-right[
.question[
  - Q. What is the mean of the distribution?
  - Q. What is the standard deviation of the distribution?]
]

--
<img src="data:image/png;base64,#../img/norms.png" width="55%" style="display: block; margin: auto;" />
---

# Worked Example

.pull-right[
- Sketch the Normal density curve for this distribution.
  - Q. What percent of ITBS scores is between 3.74 and 9.94?
  - Q. What percent of the scores is below 3.74?
]

--
<img src="data:image/png;base64,#../img/norms.png" width="55%" style="display: block; margin: auto;" />

---
.question[Check your understanding: What percent of the scores is above 5.29?]

---
# Standard Normal

- Normal is a model of the real world
- Not exact, but it is a facile model for many things
    - Physical features
    - Psychological features
    - Performance measures
--
    - Not all variables are normal
        - Skewed variables (e.g. income)
        - Any count variable (number of kids, mistakes on an exam)
---

# Real World Data
.pull-left[
- Many variables follow this distribution ( but not all)
- I have plotted histograms of data,
    - we have already used in this class
    - overlaid with the standard normal.
]
--
.small.pull-right[
<img src="data:image/png;base64,#rnorms_files/figure-html/example-1.png" width="90%" style="display: block; margin: auto;" />
]
---

# Height of Children (Galton dataset)
.small[

``` r
library(HistData)
library(ggplot2)
library(tidyverse)
Galton %>%
ggplot(aes(x = child)) + 
	geom_histogram(fill = "red") + 
	stat_function(
		fun = function(x, mean, sd, n){
			n * dnorm(x = x, mean = mean, sd = sd)
		}, 
		args = with(Galton, c(mean = mean(child), sd = sd(child), n
= length(child)))
	) + 
	scale_x_continuous("Heights of Children") + theme_minimal()
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-9-1.png" width="40%" style="display: block; margin: auto;" />
]

---

# IMBD movie ratings (movies dataset)
.pull-left.midi[

``` r
library(ggplot2movies)
data(movies)

ggmovie <- ggplot(movies, 
               aes(x = rating)) + 
	geom_histogram(fill = "blue") +
	geom_freqpoly(aes(
	  x = rnorm(length(rating))*sd(rating) + mean(rating)),
	              fill = "black")  +
	scale_x_continuous("IMBD Movie Ratings") + 
  theme_minimal()
```
]
.pull-right[
<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-10-1.png" width="90%" style="display: block; margin: auto;" />
]
---

# Temperature in Nottingham (nottem dataset)

---

# Standard Normal
- Normal distribution tricks
    - Symmetric
    - 50% of area above zero
    - Total proportion is 1.0 (or 100%)
    
---
# Area under the Normal Distribution

---
class: middle

# Wrapping Up...

---
class: middle

# Rescaling

---

# Rescaling

- All Normal distributions are the same 
  - if we measure in units of size `$\sigma$` from the mean `$\mu$` as center.
- We can convert any variable into the same metric as the standard normal
- Changing to these units is called standardizing or rescaling.

---

# Converting Formulas

.pull-left[
- Statistics Sample
- `$z_{i}$` = `$\frac{x_{i}-\bar{x}}{s}$`
]

--
.pull-right[
- Population
- `$z_{i}$` = `$\frac{x_{i}-\mu}{\sigma}$`
]

---

# Merits
.pull-left[
- Advantages
    - Allows us to compare scores on a common metric 
    - Origin is 0. The mean
        - The units are 1, the standard deviation
        - '+' values above the mean
        - '-' values below the mean
]

.pull-right[
- We can compare across measurement scales
    - Shape of the distribution does NOT CHANGE
- We can go from z-scores to raw scores
]

---

# Demo

.pull-left[
.small[

``` r
library(ggplot2movies)

# Raw data
movies$rating[1:10]
```

```
##  [1] 6.4 6.0 8.2 8.2 3.4 4.3 5.3 6.7 6.6 6.0
```

``` r
# Rescaling
variable <- movies$rating
scale(variable)[1:10]
```

```
##  [1]  0.30079877  0.04323788  1.45982279  1.45982279 -1.63090793
##  [6] -1.05139592 -0.40749368  0.49396944  0.42957922  0.04323788
```
]

- Mean = 5.93 (SD  = 1.55)
]

.pull-right[

.midi[

| Raw| Z_Score|
|---:|-------:|
| 6.4|    0.30|
| 6.0|    0.04|
| 8.2|    1.46|
| 8.2|    1.46|
| 3.4|   -1.63|
| 4.3|   -1.05|
| 5.3|   -0.41|
| 6.7|    0.49|
| 6.6|    0.43|
| 6.0|    0.04|
]
]

---

# Demo

.pull-left[

``` r
plot(density(variable)) # no scaling
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-14-1.png" width="90%" style="display: block; margin: auto;" />
]

.pull-right[

``` r
plot(density(scale(variable))) # with scaling
```

<img src="data:image/png;base64,#rnorms_files/figure-html/unnamed-chunk-15-1.png" width="90%" style="display: block; margin: auto;" />
]

---
# Worked Z-Score Problem

- Here are the IQ test scores of 31 7th-grade girls in a Midwest school district. 
<br>

---
# Worked Z-Score Problem

A) We expect IQ scores to be approximately Normal.
--

- Make a stem plot to check that there are no major departures from normality.

<img src="data:image/png;base64,#../img/iqstem.png" width="65%" style="display: block; margin: auto;" />
---

# Worked Z-Score Problem

B) Find the mean and standard deviation
--

- Mean =105.84 = `$\sum \frac{X_{i}}{n}$` = 3281/31
- SD = 14.27 =  `$s^{2}$` = `$\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})^{2}}{n-1}$` =  `$s^{2}$` = `$\frac{\sum^{n}_{i=1}(x_{i}-105.84)^{2}}{30}$`

---

# Worked Z-Score Problem

C) What proportion of scores are within one standard deviation of the mean?
- One SD above mean = 105.84 + 14.27 = 120.11
- One SD below mean = 105.84 - 14.27 = 91.57
- 23/31 = 0.74
--

---
# Worked Z-score Problem

B) What proportion of scores are within TWO standard deviations of the mean?
- TWO SD above mean = 105.84 + 2*(14.27) = 134.38
- TWO SD below mean = 105.84 - 2*(14.27) = 77.3
- 29/31 = 0.935

---

# Worked Z-score Problem

B) What would these proportions be in an exactly Normal distribution?

-  +/- One SD?

---

class:middle
# Continued in Power Point...